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## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -13, 23])

gp: E = ellinit([0, 1, 0, -13, 23])

magma: E := EllipticCurve([0, 1, 0, -13, 23]);

$$y^2=x^3+x^2-13x+23$$ ## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-1, 6\right)$$ $$\hat{h}(P)$$ ≈ $0.15999928102519923356871021721$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-1,\pm 6)$$, $$(2,\pm 3)$$, $$(7,\pm 18)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$1200$$ = $$2^{4} \cdot 3 \cdot 5^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-172800$$ = $$-1 \cdot 2^{8} \cdot 3^{3} \cdot 5^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{40960}{27}$$ = $$-1 \cdot 2^{13} \cdot 3^{-3} \cdot 5$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$-0.29408928340257600737892216666\dots$$ Stable Faltings height: $$-1.0244270558482229427572034698\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.15999928102519923356871021721\dots$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$2.9680890221372965189393751923\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$6$$  = $$2\cdot3\cdot1$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q + q^{3} - q^{7} + q^{9} - 6q^{11} - 5q^{13} + 6q^{17} - 5q^{19} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 144 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar/factorial(ar)

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L'(E,1)$$ ≈ $$2.8493526573645245690139921528667882307$$

## Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$2$$ $$I_0^{*}$$ Additive -1 4 8 0
$$3$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$5$$ $$1$$ $$II$$ Additive 1 2 2 0

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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

prime Image of Galois representation
$$3$$ B

## $p$-adic data

### $p$-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 add split add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary - 2 - 1 1 1 3 1 1 1 1 1 1,1 1 1 - 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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 1200o consists of 2 curves linked by isogenies of degree 3.

## 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 $K$ $2$ $$\Q(\sqrt{-5})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.300.1 $$\Z/2\Z$$ Not in database $6$ 6.0.270000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.21600000.1 $$\Z/6\Z$$ Not in database $6$ 6.0.7200000.1 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ 12.0.466560000000000.4 $$\Z/6\Z \times \Z/6\Z$$ Not in database $18$ 18.0.105416259632460288000000000000000.6 $$\Z/9\Z$$ 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.