Properties

Label 75.a1
Conductor $75$
Discriminant $-29296875$
j-invariant \( -\frac{102400}{3} \)
CM no
Rank $0$
Torsion structure trivial

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -208, -1256])
 
gp: E = ellinit([0, 1, 1, -208, -1256])
 
magma: E := EllipticCurve([0, 1, 1, -208, -1256]);
 

\(y^2+y=x^3+x^2-208x-1256\)  Toggle raw display

Mordell-Weil group structure

trivial

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 75 \)  =  $3 \cdot 5^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-29296875 $  =  $-1 \cdot 3 \cdot 5^{10} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{102400}{3} \)  =  $-1 \cdot 2^{12} \cdot 3^{-1} \cdot 5^{2}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.21162062212237737261518773535\dots$
Stable Faltings height: $-1.1295776382393729395521117090\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.62723492949639230259155163689\dots$
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: $ 1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $1$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Special value: $ L(E,1) $ ≈ $ 0.62723492949639230259155163689191200894 $

Modular invariants

Modular form   75.2.a.a

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

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 30
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$5$ $1$ $II^{*}$ Additive 1 2 10 0

Galois representations

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

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$5$ 5B.1.2 5.24.0.3

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ 2 3 5
Reduction type ss split add
$\lambda$-invariant(s) 0,1 1 -
$\mu$-invariant(s) 0,0 0 -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 5.
Its isogeny class 75.a consists of 2 curves linked by isogenies of degree 5.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.300.1 \(\Z/2\Z\) Not in database
$4$ \(\Q(\zeta_{5})\) \(\Z/5\Z\) Not in database
$5$ 5.1.253125.1 \(\Z/5\Z\) Not in database
$6$ 6.0.270000.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$8$ 8.2.2767921875.1 \(\Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$12$ 12.0.1012500000000.1 \(\Z/10\Z\) Not in database
$15$ 15.1.1245564843750000000000.1 \(\Z/10\Z\) Not in database
$20$ 20.0.513156902790069580078125.1 \(\Z/5\Z \times \Z/5\Z\) Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.