# Properties

 Label 245.a1 Conductor $245$ Discriminant $-42875$ j-invariant $$-\frac{110592}{125}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, -7, 12])

gp: E = ellinit([0, 0, 1, -7, 12])

magma: E := EllipticCurve([0, 0, 1, -7, 12]);

$$y^2+y=x^3-7x+12$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(7, 17\right)$$ $\hat{h}(P)$ ≈ $0.032236886628249108260991728075$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-3, 2\right)$$, $$\left(-3, -3\right)$$, $$\left(0, 3\right)$$, $$\left(0, -4\right)$$, $$\left(1, 2\right)$$, $$\left(1, -3\right)$$, $$\left(2, 2\right)$$, $$\left(2, -3\right)$$, $$\left(7, 17\right)$$, $$\left(7, -18\right)$$, $$\left(22, 102\right)$$, $$\left(22, -103\right)$$, $$\left(35, 206\right)$$, $$\left(35, -207\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$245$$ = $5 \cdot 7^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-42875$ = $-1 \cdot 5^{3} \cdot 7^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{110592}{125}$$ = $-1 \cdot 2^{12} \cdot 3^{3} \cdot 5^{-3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.41676082073262685941866133847\dots$ Stable Faltings height: $-0.90323835799645518569499952433\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.032236886628249108260991728075\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $3.2744322569434663260680266853\dots$ 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: $6$  = $3\cdot2$ 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: $L'(E,1)$ ≈ $0.63334500863381026730431838091486857774$

## Modular invariants

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 48 $\Gamma_0(N)$-optimal: yes 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$(j)_{-}$
$5$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$7$ $2$ $III$ Additive -1 2 3 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
$3$ 3Nn 3.3.0.1

## $p$-adic regulators

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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

## Iwasawa invariants

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

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

## Isogenies

This curve has no rational isogenies. Its isogeny class 245.a consists of this curve only.

## 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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.140.1 $$\Z/2\Z$$ Not in database $6$ 6.0.686000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.257298363.1 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $16$ 16.0.66202447602479769.1 $$\Z/3\Z \times \Z/3\Z$$ 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.