Properties

Label 30030bt1
Conductor 30030
Discriminant -3157693080314572800
j-invariant \( -\frac{46555485820017544148689}{3157693080314572800} \)
CM no
Rank 1
Torsion Structure \(\Z/{12}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -749461, 263897441]); // or
magma: E := EllipticCurve("30030bt1");
sage: E = EllipticCurve([1, 0, 0, -749461, 263897441]) # or
sage: E = EllipticCurve("30030bt1")
gp: E = ellinit([1, 0, 0, -749461, 263897441]) \\ or
gp: E = ellinit("30030bt1")

\( y^2 + x y = x^{3} - 749461 x + 263897441 \)

Mordell-Weil group structure

\(\Z\times \Z/{12}\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(-880, 16001\right) \)
\(\hat{h}(P)\) ≈  1.63505616079

Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

\( \left(-334, 22007\right) \)

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

\( \left(-1006, 503\right) \), \( \left(-880, 16001\right) \), \( \left(-334, 22007\right) \), \( \left(-34, 17027\right) \), \( \left(128, 12977\right) \), \( \left(290, 8279\right) \), \( \left(394, 5263\right) \), \( \left(446, 4067\right) \), \( \left(506, 3527\right) \), \( \left(758, 11087\right) \), \( \left(1010, 22679\right) \), \( \left(1346, 40487\right) \), \( \left(2396, 109367\right) \), \( \left(5906, 446327\right) \), \( \left(17138, 2232215\right) \), \( \left(34316, 6337757\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 30030 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(-3157693080314572800 \)  =  \(-1 \cdot 2^{12} \cdot 3^{12} \cdot 5^{2} \cdot 7^{4} \cdot 11 \cdot 13^{3} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( -\frac{46555485820017544148689}{3157693080314572800} \)  =  \(-1 \cdot 2^{-12} \cdot 3^{-12} \cdot 5^{-2} \cdot 7^{-4} \cdot 11^{-1} \cdot 13^{-3} \cdot 47^{3} \cdot 59^{3} \cdot 12973^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
Rank: \(1\)
magma: Regulator(E);
sage: E.regulator()
Regulator: \(1.63505616079\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(0.248157514001\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
Tamagawa product: \( 3456 \)  = \( ( 2^{2} \cdot 3 )\cdot( 2^{2} \cdot 3 )\cdot2\cdot2^{2}\cdot1\cdot3 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(12\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 30030.2.a.bt

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - q^{11} + q^{12} + q^{13} + q^{14} - q^{15} + q^{16} - 6q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
774144 . This curve is \( \Gamma_0(N) \)-optimal.

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])

\( L'(E,1) \) ≈ \( 9.73803533073 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(12\) \( I_{12} \) Split multiplicative -1 1 12 12
\(3\) \(12\) \( I_{12} \) Split multiplicative -1 1 12 12
\(5\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(7\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(11\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(13\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X13h.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 3 \end{array}\right)$ and has index 12.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(2\) B
\(3\) B.1.1

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split split nonsplit split nonsplit split ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 4 2 1 2 1 2 1 1 1,1 1 1 1 1 1 1,1
$\mu$-invariant(s) 0 0 0 0 0 0 0 0 0,0 0 0 0 0 0 0,0

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 30030.bt consists of 8 curves linked by isogenies of degrees dividing 12.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{12}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{-143}) \) \(\Z/2\Z \times \Z/12\Z\) Not in database
4 4.2.100900800.11 \(\Z/24\Z\) Not in database
6 6.0.593207566875.1 \(\Z/3\Z \times \Z/12\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.